Could a class A stellar engine (or maybe a class C one) be built using a brown dwarf (for argument's sake, a T-dwarf like Gliese 229B)? Would it be capable of enough thrust to move itself any appreciable distance over a long enough period of time? (Let's say over a period of a billion years.)

As a possible follow-up if the answer above is more or less "yes", would any appreciable additional thrust be possible to create by generating an asymmetrical solar-flare like eruption in a modified variation of a thermal-driven outflow "star lifting" technique?

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The basis for a stellar engine (according to the Wikipedia link) is that the radiation pressure from the star is used to create thrust, so the question reduces to "Do brown dwarfs emit enough radiation"? Stars like the Sun do emit radiation, and lots of it - in the Sun's case, $3.846 \times 10^{26}$ watts. That's a lot of power! Brown dwarfs can't undergo hydrogen fusion and enter the main sequence, emitting light like the Sun, but they may emit X-rays (https://en.wikipedia.org/wiki/Brown_dwarf#Brown_dwarf_as_an_X-ray_source). You would need a lot of X-rays to give the stellar engine any appreciable power, and while I can't find that data to prove it, it seems like brown dwarfs would be poor power sources for stellar engines.